パーセバルの等式

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パーセバルの等式

 

パーセバルの等式とは

\(f(x)\)のフーリエ級数の係数に関して、以下がパーセバルの等式です。

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} {f(x)}^2 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)

 

証明

\(f(x)=\displaystyle\frac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)\)

 

というフーリエ級数式を使って計算する。

 

\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x)\cdot f(x) dx\)

 

\(=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x)\cdot \biggl[\displaystyle\frac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)\biggr] dx\)

 

\(=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\frac{a_{0}}{2}f(x) dx+\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\sum_{n=1}^{\infty}a_{n}f(x)\cos nx dx+\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\sum_{n=1}^{\infty}b_{n}f(x)\sin nx dx\)

 

\(=\displaystyle\frac{a_{0}}{2}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) dx\)\(+\displaystyle\sum_{n=1}^{\infty}a_{n}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \cos nx dx\)\(+\displaystyle\sum_{n=1}^{\infty}b_{n}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \sin nx dx\)

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) dx=a_{0}\)

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \cos nx dx=a_{n}\)

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}f(x) \sin nx dx=b_{n}\)

 

を代入すると、

 

\(=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)=右辺\) となる。

 

例題

1番

\(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^2}=\displaystyle\frac{\pi^2}{6}\)   を示す。

 

\(f(x)=x\)  とおく。奇関数なので、\(a_{0}\)、\(a_{n}\)は0。\(b_{n}\)を部分積分で求めると

 

\(b_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x\sin nx dx=\displaystyle\frac{2}{n}(-1)^{n-1}\)

 

これらをパーセバルの等式に代入する。

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}x^2 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)

 

\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2 dx=\displaystyle\frac{2}{3}\pi^2\)

 

\(右辺=\displaystyle\sum_{n=1}^{\infty}\biggl[\displaystyle\frac{2}{n}(-1)^{n-1}\biggr]^2=\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{4}{n^2}\)

 

よって \(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^2}=\displaystyle\frac{\pi^2}{6}\)

 

2番

\(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^4}=\displaystyle\frac{\pi^4}{90}\)   を示す。

 

\(f(x)=x^2\)  とおく。偶関数なので、\(b_{n}\)は0。\(a_{0}\)と\(a_{n}\)をそれぞれ求めると

 

\(a_{0}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2 dx=\displaystyle\frac{2}{3}\pi^2\)

 

\(a_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2\cos nx dx=\displaystyle\frac{4}{n^2}(-1)^n\)

 

これらをパーセバルの等式に代入する。

 

\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}x^4 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)

 

\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^4 dx=\displaystyle\frac{2}{5}\pi^4\)

 

\(右辺=\displaystyle\frac{2}{9}\pi^2+\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{16}{n^4}\)

 

よって \(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^4}=\displaystyle\frac{1}{16}\biggl(\displaystyle\frac{2}{5}\pi^4-\displaystyle\frac{2}{9}\pi^4\biggr)=\displaystyle\frac{\pi^4}{90}\)

 

 

 

 

 

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